First of all, we recall that,
according to the general theory of elliptic problems in domains with
cylindrical outlets to infinity, see [1]} and [2]}, the homogeneous problem (REF ), (REF ), (REF ) has
just two\(2=\frac{1}{2}(2\times 2)\)  where the last 2 is
the number of outlets to infinity in the domain \(\Xi \)  and the next
to the last 2 is the number of linearly independent, \(H\) -periodic
in \(\xi _2\)  and polynomial in \(\xi _1\) , harmonics in the intact strip
\(\Pi \) , namely 1 and \(\xi _1\)  in our case. This mnemonic rule works
for many other problems in domains with cylindrical and periodic
outlets to infinity, see the review paper [2]}.
 linearly independent solutions with the polynomial behavior at infinity.  It is evident that the first solution is a constant, and we set
\(W^0(\xi )=1.\) 
